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Shape recognition via Wasserstein distance
Wilfrid Gangbo and Robert J. McCann
The Kantorovich-Rubinstein-Wasserstein metric defines the distance between two probability measures f and g on Rd+1 by computing the cheapest way to transport the mass of f onto g, where the cost per unit mass transported is a given function c(x,y) on R2d+2. Motivated by applications to shape recognition, we analyze this transportation problem with the cost c(x,y) = |x-y|2 and measures supported on two curves in the plane, or more generally on the boundaries of two domains U and V in Rd+1. Unlike the theory for measures which are absolutely continuous with respect to Lebesgue, it turns out not to be the case that f-a.e. boundary point of U is transported to a single image y on the boundary of V; however, we show the images of x are almost surely collinear and parallel the normal to U at x. If either domain is strictly convex, we deduce that the solution to the optimization problem is unique. When both domains are uniformly convex, we prove a regularity result showing the images of x are always collinear, and both images depend on x in a continuous and (continuously) invertible way. This produces some unusual extremal doubly stochastic measures.